Karnaugh Maps

The Karnaugh map (K–map), introduced by Maurice Karnaughin in 1953, is a grid-like representation of a truth table which is used to simplify boolean algebra expressions. A Karnaugh map has zero and one entries at different positions. It provides grouping together Boolean expressions with common factors and eliminates unwanted variables from the expression. In a K-map, crossing a vertical or horizontal cell boundary is always a change of only one variable.

Example 1

An arbitrary truth table is taken below −

A

B

A operation B

0

0

w

0

1

x

1

0

y

1

1

z

Now we will make a k-map for the above truth table −

Example 2

Now we will make a K-map for the expression − AB+ A’B’

Simplification Using K-map

K-map uses some rules for the simplification of Boolean expressions by combining together adjacent cells into single term. The rules are described below −

Rule 1 − Any cell containing a zero cannot be grouped.

Wrong grouping

Rule 2 − Groups must contain 2n cells (n starting from 1).

Wrong grouping

Rule 3 − Grouping must be horizontal or vertical, but must not be diagonal.

Wrong diagonal grouping

Proper vertical grouping

Proper horizontal grouping

Rule 4 − Groups must be covered as largely as possible.

Insufficient grouping

Proper grouping

Rule 5 − If 1 of any cell cannot be grouped with any other cell, it will act as a group itself.

Proper grouping

Rule 6 − Groups may overlap but there should be as few groups as possible.

Proper grouping

Rule 7 − The leftmost cell/cells can be grouped with the rightmost cell/cells and the topmost cell/cells can be grouped with the bottommost cell/cells.

Proper grouping

Problem

Minimize the following Boolean expression using K-map −

$$F (A, B, C) = A'BC + A'BC' + AB'C'+ AB'C$$

Solution

Each term is put into k-map and we get the following −

K-map for F (A, B, C)

Now we will group the cells of 1 according to the rules stated above −

K-map for F (A, B, C)

We have got two groups which are termed as $A’B$ and $AB’$. Hence, $F (A, B, C) = A’B+ AB’= A \oplus B$. It is the minimized form.